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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 7

Question

Math 6010 • Linear Models Fall 2017 • Question 7

Let \(X_1,X_2,\ldots ,X_n\) be iid with density \[ f(x)=2x/\theta ^2, \qquad 0<x<\theta . \] Let \(Y=X_{(n)}=X_{n:n}\) be the largest order statistic of the \(X_i\). Show that \(Y/\theta \) is a pivotal quantity for \(\theta \) and use it to construct a 99% confidence interval for \(\theta \).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be iid with density
\[
  f(x)=2x/\theta^2, \qquad 0<x<\theta.
\]
Let \(Y=X_{(n)}=X_{n:n}\) be the largest order statistic of the \(X_i\). Show that
\(Y/\theta\) is a pivotal quantity for \(\theta\) and use it to construct a 99\%
confidence interval for \(\theta\).

Topic Tags

Estimator Distributions Measure-Theoretic Probability Simultaneous Confidence Intervals Uniform Integrability and Truncation Methods